Page 36 - IJOCTA-15-4

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.4, pp.578-593 (2025)
https://doi.org/10.36922/IJOCTA025110050

RESEARCH ARTICLE

Nonlinear image processing with α-tension field: A geometric
approach

2
1
Seyyed Mehdi Kazemi Torbaghan , Yaser Jouybari Moghaddam , and Amin Jajarmi 3,4*
1
Department of Mathematics, University of Bojnord, Bojnord, Iran
2
Department of Surveying Engineering, University of Bojnord, Bojnord, Iran
3
Department of Electrical Engineering, University of Bojnord, Bojnord, Iran
4
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical
Sciences, Saveetha University, Chennai,Tamil Nadu, India
m.kazemi@ub.ac.ir, a.jajarmi@ub.ac.ir, jouybari@ub.ac.ir
ARTICLE INFO ABSTRACT
Article History:
Received: March 13, 2025
In this paper, we apply an α-tension field from differential geometry to classi-
1st revised: March 27, 2025
2nd revised: May 3, 2025 cal image processing tasks of denoising with edge preservation and multiphase
3rd revised: May 16, 2025 feature enhancement. The main contribution of this work is that it is the first
Accepted: May 22, 2025 systematic investigation of the α-tension field for image processing. Contrary
Published Online: June 6, 2025 to traditional operators, such as the Laplacian, which are susceptible to noise
amplification or are ineffective for complex structures, the α-tension field re-
Keywords: lies on a nonlinear adaptive mechanism depending on the magnitudes of local
Image processing gradients. It allows effective denoising and retains edges and fine details by uti-
α−tension field lizing higher-order gradient information. The field of α-tension provides more
Harmonic maps sensitive and adaptive models than linear models, such as total variation regu-
Riemannian geometry larization, anisotropic diffusion, etc. The study exemplifies its advantages over
Subject Classification: previous methods in preserving structural integrity and minimizing artifacts.
68U10;53C43; 58E20, It also considers numerical implementation issues and provides guidelines for
real-time and large-scale processing. This framework adds up to the known
need for faster image-processing tools while links connections to differential
geometry.

1. Introduction and background harmonic maps. To tackle this problem, Karen
Uhlenbeck and Jonathan Sacks introduced the
The tension field is one of the important ana- α-tension field, which corresponds with the α-
2
lytic and geometric objects to study in the con- energy functional E α (u) = R M (1+ | du | 2
α
text of harmonic mappings between Riemannian ) dvol g , for α > 1. This approach guarantees
manifolds. For a smooth map u : (M, g) −→ connectivity, and it is used to build geomet-
(N, h), we define the tension field of u to be ric objects called α-harmonic maps which are
the trace of the second fundamental form of u, defined as the maps with vanishing α-tension
so that τ(u) = traceg∇du, where ∇ is the in- field. These α-harmonic maps are used to con-
1
duced connection on u −1 TN. Harmonic maps struct the standard harmonic maps in the limit
are, therefore, critical points of the energy func- α → 1. 2
tional, and they satisfy the Euler-Lagrange equa-
tion τ(u) = 0. Nevertheless, the non-coercivity The work of Karen Uhlenbeck on α−tension
of the standard energy functional frequently adds field addresses fundamental questions regarding
technical difficulties to the existence theory for the existence and regularity of harmonic maps
*Corresponding Author
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